AH of the fame given magnitude which can be drawn from a given point A to a straight line BC given in position. PROP. XXXV. IF a straight line be drawn between two parallel straight lines given in pofition, and makes given angles with them; the straight line is given in magnitude. Let the ftraight line EF be drawn between the parallels AB, CD which are given in pofition, and make the given angles BEF, EFD; EF is given in magnitude. A EH B 32. b. 29. I. In CD take the given point G, and thro' G draw a GH paral- a. 31. 1. lel to EF. and because CD meets the parallels GH, EF, the angle EFD is equal b to the angle HGD. and EFD is a given angle, wherefore the angle HGD is given. and because HG is drawn to the given point G in the ftraight line CD given in position, and makes a given angle HGD; the ftraight line HG is given in pofition . and AB is given in pofition, therefore the point c. 32. Dat. H is given ; and the point G is also given, wherefore GH is given in magnitude and EF is equal to it; therefore EF is given in e. 29. Dat. magnitude. C F G D C. d. 28. Dat. IF F a straight line given in magnitude be drawn between See N. two parallel ftraight lines given in pofition; it fhall make given angles with the parallels. Let the ftraight line EF given in magnitude be drawn between the parallel ftraight lines AB, CD which are given in pofition; the angles AEF, EFC shall be given. A EHB Because EF is given in magnitude, a ftraight line equal to it can be found a; let C this be G. in AB take a given point H, and G from it draw HK perpendicular to CD. b therefore the straight line G, that is EF, cannot be less than HK. and if G be equal to HK, EF alfo is equal to it; wherefore EF is at right angles to CD, for if it be not, EF would be greater than HK, which is abfurd. therefore the angle EFD is a right and confequently a given angle. But if the ftraight line G be not equal to HK, it must be greater than it. produce HK, and take HL equal to G; and from the d center H, at the distance HL describe the circle MLN, and join c. 6. Def. HM, HN. and because the circle MLN, and the straight line CD d. 28. Dat. are given in pofition, the points M, N are given; and the point H is given, wherefore the straight lines HM, HN are given in pofi-A e. 29. Dat. tion e. and CD is given in position, therefore the angles HMN, g. 34. I. H B K C F OML ND G f. A. Def. HNM are given in pofition . of the ftraight lines HM, HN let HN be that which is not parallel to EF, for EF cannot be parallel to both of them; and draw EO parallel to HN. EO therefore is equal 8 to HN, that is to G; and EF is equal to G, wherefore EO is equal to EF, and the angle EFO to the angle EOF, that is to the given angle HNM. and because the angle HNM which is equal to the angle EFO or EFD has been found, therefore the angle EFD, that is the angle AEF, is given in magk. 1. Def. nitude 3, and confequently the angle EFC. h. 29. I. See N. E. IF a PROP. XXXVII. Fa ftraight line given in magnitude be drawn from a point to a ftraight line given in pofition, in a given angle; the ftraight line drawn thro' that point parallel to the straight line given in pofition, is given in pofition.. Let the ftraight line AD given in magnitude be drawn from the point A to the ftraight line BC given in pofition, in the given angle ADC; the ftraight E line EAF drawn through A parallel to BC is given in pofition. AHE In BC take a given point G, and draw GHB D G. C parallel to AD. and because HG is drawn to a given point G in the straight line BC given in position, in a given angle HGC, for it is equal to the given angle ADC; HG is a. 29. 1. given in pofition ; but it is given alfo in magnitude, because it is b. 32. Dat. equal to AD which is given in magnitude. therefore because G one of the extremities of the straight line GH given in pofition and magnitude is given, the other extremity H is given . and the c. 30. Dat. straight line EAF which is drawn through the given point H parallel to BC given in pofition, is therefore given in position. d. 31. Dat. PROP. XXXVIII. d a ftraight line be drawn from a given point to two parallel straight lines given in pofition; the ratio of the fegments between the given point and the parallels shall be given. Let the ftraight line EFG be drawn from the given point E to the parallels AB, CD; the ratio of EF to EG is given. From the point E draw EHK perpendicular to CD. and because from a given point E the ftraight line EK is drawn to CD which is given in position, in a given angle EKC; EK is given in pofi 34. b. 28. Dat. c. 29. Dat. tion *. and AB, CD are given in pofition; therefore the points a. 33. Dat. H, K are given. and the point E is given, wherefore EH, EK are given in magnitude, and the ratio of them is therefore given. d. 1. Dat. but as EH to EK, fo is EF to EG, becaufe AB, CD are parallels. therefore the ratio of EF to EG is given. PROP. XXXIX. 35.36. If ratio the of a F the ratio of the fegments of a straight line between See N. a given point in it and two parallel ftraight lines be given; if one of the parallels be given in pofition, the other is alfo given in pofition. B b From the given point A let the straight line AED be drawn to the two parallel ftraight lines FG, BC, and let the ratio of the segments AE, AD be given; if one of the parallels BC be given in pofition, the other FG is alfo given in pofition. From the point A draw AH perpendicular to BC, and let it meet FG in K. and because AH is drawn from the given point A to the ftraight line BC given in pofition, and makes a given angle a a. 33. Dat. AHD; AH is given a in position. and BC is likewife given in pofition, there b. 28. Dat. fore the point H is given. the point A B is alfo given, wherefore AH is given in FE D/H C c. 29. Dat. magnitude . and, because FG, BC are parallels, as AE to AD, fo is AK toK G AH; and the ratio of AE to AD is given, wherefore the ratio of AK to AH is given; but AH is given d. 2. Dat. in magnitude, therefore a AK is given in magnitude; and it is also e. 30. Dat. given in pofition, and the point A is given; wherefore the point K is given. and because the straight line FG is drawn thro' the 31. Dat. given point K parallel to BC which is given in position, therefore £ FG is given in pofition. 37.38. See N. PROP. XL. F the ratio of the fegments of a straight line into which it is cut by three parallel ftraight lines, be given; If two of the parallels are given in pofition, the third alfo is given in pofition. Let AB, CD, HK be three parallel ftraight lines, of which AB, CD are given in pofition; and let the ratio of the fegments GE, GF into which the straight line GEF is cut by the three parallels, be given; the third parallel HK is given in pofition. In AB take a given point L, and draw LM perpendicular to CD, meeting HK in N. because LM is drawn from the given point L to CD which is given in pofition, and makes a given angle LMD; LM is given in pofitiona. and CD is given in position, a. 33. Dat. wherefore the point M is given; and the point L is given, LM b. 28. Dat. is therefore given in magnitude . and because the ratio of GE to c. 29. Dat. d Cor. 6. or 17.Dat. e. 2. Dat. GF is given, and as GE to GF, fo is NL to NM; the ratio of NL to NM is given; and therefore the ratio of ML to LN is d. given. but LM is given in magnitude, wherefore e LN is given in magnitude; and it is also given in position, and the point L is given; wherefore the point N is given. and because the straight f. 30. Dat. line HK is drawn thro' the given point N parallel to CD which is given in pofition, therefore HK is given in position 8. f PROP. XLI. g. 31. Dat. F. If meets IF Fa ftraight line meets three parallel ftraight lines See N. which are given in pofition; the fegments into which they cut it, have a given ratio. Let the parallel straight lines AB, CD, EF given in position be LM perpendicular to CD, meeting EF in A and the ratio of LM to MN is therefore G L B ati M D b. 29, Dat. N F |